# Mathematics

As a high school mathematics teacher, I usually introduce relations and functions for the first time to ninth grade algebra students and throughout their mathematical experience, my older secondary students will continue to learn more about functions as they advance. A recent interaction I had with my daughter, who is currently in second grade, got me thinking: what “secondary” math concepts can be introduced at elementary levels? Not too long ago, I was at home working on a lesson for my precalculus class involving transformations of parent functions. My young daughter was watching what I was doing on the computer and after standing next to me for a few minutes, she asked: “Tata, what’s a function?”

I thought about what I was going to say for a moment, then I replied, “a fun rule or instruction”. She looked a little surprised about the “fun” part so she gave me a bit of a thinking look so I stopped what I was doing and asked her to get the whiteboards and markers. I told her a function was a “fun” rule that took something to make something new. Now, she was even more interested and wanted to know how this FUNction worked. I thought about using a rule involving addition and subtraction (the two operations she’s most familiar with) but I felt she was expecting something a little more “fun”.

I decided to use our geometry shapes set and I told her we were going to make a function for the triangle. I asked her what do you think we can make with a triangle? She replied, can we make another triangle? At that point, we had our “fun rule or instruction”: use the triangle to make another triangle. We grabbed a handful of green triangles and played around to make a bigger green triangle consisting of smaller triangles.

At this point, we had defined a more precise fun rule: “make a bigger shape that looks like the original shape”. We then removed the triangles from the whiteboard and started using other shapes we had available. This time, my daughter oversaw choosing the shape and applying our function.

After a few examples I asked her, what do you notice about the bigger shape? She replied it has the same color and always has four of the smaller shapes. Our “fun” function also worked with a rhombus but not with a hexagon. I really enjoyed our exploration and I felt like my daughter had gained valuable insights into the concept of a function. Overall, it got me thinking: what are other opportunities to explore functions with younger students? I immediately thought about one of my favorite activations: Fawn Nguyen’s visual patterns. I have used them with my students for different purposes and recently with my daughter as she was starting to learn multiplication. I showed my daughter the pattern below and asked her to predict step four.

After some observation and writing down some ideas, she figured out the next step will involve a domino with four dots on top and five dots on the bottom. However, she wasn’t sure about the dice. She knew they should add up to twenty but did not know if there was a specific way she needed to organize the sum. We did not mention the concept of function when we were exploring the visual pattern, but she was able to predict the next output based on previous input/output values. She was analyzing a function while working with the pattern.

Thus far, my fourteen years of teaching experience has been at the secondary level so I have only experienced teaching elementary concepts to my own daughter at home. However, this momentary experience has made me wonder what other “secondary” concepts can be introduced at the elementary levels. I’d love to learn more about it as I think it could be very powerful for students of different ages to realize they’re learning the same mathematical ideas.

I really hope you find these ideas helpful and can use them with your students in the near future. I would love to learn about other “secondary” concepts math teachers have introduced to younger students. What was your experience like? How did your students respond?